A new computational method for nonlinear normal modes of nonconservative systems

摘要

The concept of nonlinear normal modes (NNMs) was introduced with the aim of providing a rigorous generalization of normal modes to nonlinear systems. Initially, NNMs were defined as periodic solutions of the underlying conservative system, and continuation algorithms were recently exploited to compute them. To extend the concept of NNMs to nonconservative systems, Shaw and Pierre defined an NNM as a two-dimensional invariant manifold in the system’s phase space. This contribution presents a novel algorithm for solving the set of partial differential equations governing the manifold geometry numerically. The resolution strategy takes advantage of the hyperbolic nature of the equations to progressively solve them in annular regions. Each region is defined by two different iso-energy curves and equations are discretized using a specific finite element technique. The proposed strategy also offers the opportunity to estimate the frequency-energy dependence of the mode without using time integration. The algorithm is applied to both conservative and nonconservative systems.